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6
votes
0answers
231 views

Negative curvature in the middle of $R^{3}$

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside? Basically, I am asking for a ...
4
votes
0answers
143 views

Obtaining the metric from the mixed Ricci tensor $R^i{}_j$

In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$). But what do we know about ...
3
votes
0answers
71 views

Flows associated with Killing fields

Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the ...
3
votes
0answers
85 views

Growth of norm of curvature under direct sum or existence of universal connection

For a Hermitian vector bundle E over a Riemannian $(X,g)$ and a unitary connection A we may define the sup norm of the curvature by: $$\|R_A\|=\sup \{\| tracefree (R_A (v))\|_{op} \mid v \in \Lambda ...
3
votes
0answers
69 views

Classification of manifolds with ${\rm Ric}\geq 0$ wrt fundamental group

Note that $n$-manifolds $M$ with ${\rm Ric}\geq 0$ has a fundamental group of polynomial growth of degree $\leq n$ (proof : use Bishop volume theorem). (Here a group $\Gamma$ is said to have ...
3
votes
0answers
150 views

Strake's splitting theorem for complex sectional curvature

Strake's splitting theorem: Let $(M^n,g)$ be an open manifold with sectional curvature $K\ge 0$ and soul $\Sigma^k$. If the normal holonomy group of $\Sigma$ is trivial, $M$ splits isometrically into ...
2
votes
0answers
72 views

Bounding distance between geodesics in manifolds with nonpositive curvature

This is a duplicate of a question at the stackexchange which was not answered. I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I ...
1
vote
0answers
18 views

Approximation with a more regular function and an inequality constraint

The motivation of the question comes from a geometric problem: can we approximate a $C^{1,\alpha}$ set $\Omega$ with positive curvature (in distributional sense) from inside with $C^2$ sets with ...
1
vote
0answers
38 views

Normal fields of geodesic spheres

This question is related to this one (http://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I've asked at math.stackexchange. Let $(M,g)$ be a compact Riemannian ...
1
vote
0answers
72 views

Curvature tensor for a singular manifold

Given a manifold $M$ with its tangent space $TM$ and frame vector field $e \in TM$. However, the transition functions in this tangent bundle are non-smooth. Therefore, the Lie derivative of $e$ with ...
1
vote
0answers
149 views

Spherical cap is the only compact constant mean curvature surface bounded by a circle

I would like to see that the only compact rotationally invariant constant mean curvature surfaces with boundary a planar circle, are either a planar disk or a spherical cap. This is stated in the ...
1
vote
0answers
93 views

Space with $Ric \geq -(n-1)$

Note that hyperbolic space $H$ has $Ric=-(n-1)$. I want to know : Question : Does there exists a simply connected open complete Riemannian manifold $M$ s.t. (1) $ Ric\geq -(n-1)$ on $M$ (2) $ ...
1
vote
0answers
90 views

Exposition of the Calabi complex

I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature". The complex is referenced as "Calabi complex" in various citing ...
1
vote
0answers
244 views

Formula for the curvature of an induced connection

Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...
0
votes
0answers
69 views

What is the relation between two Riemannian metrics with the same Riemannian curvatures?

Consider two flat Riemannian metrics on a manifold. What is the general relation between these two metrics if the manifold is not simply-connected? What is the answer if two Riemannian metrics have ...
0
votes
0answers
262 views

Integral of Square of Mean Curvature

Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus. When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and ...
0
votes
0answers
150 views

General form of a metric affine connection with zero curvature

I have read somewhere that the general form of a metric affine connection whose Riemann curvature is zero is given by $$\Gamma^{i}_{ij}=A^{i}_{\alpha}\partial_{j}A^{\alpha}_{k}$$ where ...